# Macroeconomic Analysis Homework 1

Over recent years, suppose real GDP of the US (Country A) has been growing at an average rate of 2% per year. Another Country B’s real GDP is about 30% of the US real GDP in current year. If the current real GDP of the US is 100 units, then real GDP of Country B will be 30 units. Use the data given and finish the blanks in the following table.

Unit | Growth rate | Current GDP | 5-year
GDP |
10-year GDP | 25-year GDP | 75-year GDP |

Country A | 2.0% | 100.00 | ||||

Country B | 5.0% | 30.00 | ||||

GDP ratio | 0.30 |

(1) If the US economic growth rate is 2% per year, then how much would be the US real GDP after 5 years, 10 years, 25 years and 75 years. You can use US current GDP (Country A) 100 as starting point value. Keep two or three decimal digits for your answer.

(2) Similarly, for Country B, with a long-run growth rate at 5% per year, how much would be Country B’s real GDP in the next 5 years, 10 years, 25 years and 75 years. You can use current GDP of country B 25 as starting point. Keep two or three decimal digits for your answer.

(3) Following questions above, how much is the ratio of country B’s real GDP to the US GDP (GDP* _{B}*/GDP

*) in the next 5 years, 10 years, 25 years and 75 years. Keep two or three decimal digits for your answer.*

_{A}

(4) If the US GDP grows faster than 2% in the next 25 years, how about the ratio of Country B’s GDP to Country A (US)’s GDP? It will be higher or lower?

**Question 2**.

The following table presents the Real Gross Domestic Product (GDP) of the United States (Chained 2012 dollars) from 1999 to 2020. Please finish the questions as below.

Year |
Real GDP |
Ln(GDP) |
GR-1 |
GR-2 |

1999 | 12610.50 | 9.442 | NA. |
NA. |

2000 | 13130.99 | 9.483 | 0.041 | |

2001 | 13262.08 | |||

2002 | 13493.06 | |||

2003 | 13879.13 | |||

2004 | 14406.38 | |||

2005 | 14912.51 | |||

2006 | 15338.26 | |||

2007 | 15626.03 | |||

2008 | 15604.69 | |||

2009 | 15208.83 | |||

2010 | 15598.75 | |||

2011 | 15840.66 | |||

2012 | 16197.01 | |||

2013 | 16495.37 | |||

2014 | 16912.04 | |||

2015 | 17432.17 | |||

2016 | 17730.51 | |||

2017 | 18144.11 | |||

2018 | 18687.79 | |||

2019 | 19091.66 | |||

2020 | 18384.69 |

Note: Real GDP, annual real GDP of the United States, unit: Billions of Chained 2012 Dollars, Not Seasonally Adjusted. Ln(GDP) is the natural logarithm of real GDP. GR-1 is the growth rate of real GDP calculated using the method in part (2) as below. GR-2 is the growth rate of real GDP calculated using method in part (3). Data: https://fred.stlouisfed.org/series/GDPCA

(1) Transform the real GDP (the 2nd column) into natural logarithm (Ln) values (in the 3rd column). There are two example years of 1999 and 2000. You may use calculator, spreadsheet or other devices to calculate the natural logarithm values. Please fill in the blanks of blue color in the third column.

(2) Calculate real GDP growth rate (GR-1) in each year and complete the blanks in the fourth column. You can use the formula as below. GDP growth rate in year *t* equals,

GR* _{t}* = (GDP

*– GDP*

_{t}*) / GDP*

_{t-1}*;*

_{t-1}In the table, real GDP growth rate in the year 2000, between 1999 and 2000, is given as an example (*t* = 2000). You can follow the example and fill in all the blanks in pink color. Please make sure to keep three decimal digits for your answers of GR-1 (as the example).

(3) An alternative approach, we can use the difference of natural logarithm to calculate growth rate (GR-2). It is relatively easier than the calculation above (GR-1). Here we will skip some technical details. In sum, for a small growth rate, we can use natural logarithm to calculate GDP growth rate. For GDP growth rate over one year, GR-2 can be calculated as,

GR* _{t}* = Ln(GDP

*) – Ln(GDP*

_{t}*); (*

_{t-1}*I*)

For the average growth rate of *N* years (GR* _{N}*),

GR* _{N}* = [Ln(GDP

*) – Ln(GDP*

_{t+N}*)]/*

_{t}*N*; (

*II*)

Now use formula (*I*) in this part to calculate real GDP growth rate (GR-2) of each year. Fill the blanks in the last column with green color and compare the results with annual growth rates computed by the method of GR-1. Are the values in the two columns close enough or not? Please make sure to keep three decimal digits for your answers.

(4) Use the annual real GDP growth rates from 2016 to 2020 and calculate the average growth rate over these five years. How much in the average growth rate? Please make sure to keep three decimal digits for your answer.

(5) With the average growth rate computed above, how many years it takes for the US to double its real GDP. Try to use both the “Rule of 70” and the logarithm method in your lecture note. Compare the results from these two methods, are they close or not. Please make sure to keep three decimal digits for your answer.

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**Question 3. **

Review the definitions of constant return to scale (CRS), increasing return to scale (IRS) and decreasing return to scale (DRS) to production functions in your notes. Explain the following production functions are CRS, IRS or DRS. To get full credit, you need show your work via step-by-step derivations.

(1) In country C, the production function determines national output (*Y*) is a function of capital (*K*) and labor (*L*) as below. Is the function CRS, IRS or DRS?

*Y* = A * *K** ^{(1/4)}* *

*L*

^{(2/3}

^{)};

(2) In country D, suppose in its production function, national output (*Y*) is a function of capital (*K*), labor (*L*) and exchange of information (*E*). Is the function CRS, IRS or DRS?

*Y* = A * *K** ^{(1/6)}* *

*L*

^{(1/3}

^{)}*

*E*

^{(1/2}

^{)};

**Question 4.**

The first column of the table below shows you the ratio real GDP per capita relative to the US level, across several countries. In the second column, you can see capital per capita (k) relative to the US level. Try to finish the table.

Country | GDP per capita (y) |
Capital per capita (k) |
Capital input k ^{(1/3)} |
TFP (A) |

US | 1.000 | 1.000 | ||

Switzerland | 1.147 | 1.416 | ||

UK | 0.733 | 0.833 | ||

Japan | 0.685 | 1.021 | ||

Italy | 0.671 | 1.125 | ||

Spain | 0.615 | 1.128 | ||

Brazil | 0.336 | 0.458 | ||

South Africa | 0.232 | 0.218 |

Note: Both GDP per capita and Capital per capita are the relative ratio to the levels of the US.

With the Cobb-Douglas production function we can calculate the value of national output or GDP (*Y*) with a combination of capital (*K*) and labor (*L*) inputs and total factor productivity (A).

*Y* = A * *K** ^{α}* *

*L*

^{(1-}

^{α)};

At per capita level, we can rewrite Cobb-Douglas function as below. Here *y* = *Y/L* and *k* = *K/L*. Here we have GDP per capita (*y*) and capital per capita (*k*). Also, we assume α is the same.

y = A** k** ^{α}*;

(1) Fill in the blanks (green color) of the values of capital input in the fourth column. Now you have the value of parameter α = 1/3. Keep three decimal digits for your answer.

(2) Following the above question, now use GDP per capita (*y*) and capital per capita (*k*) to derive total factor of productivity (A) of each country. Fill in the blanks of pink color in the last column.

(3) Can you derive Cobb-Douglas function at per capita level using y and k from Cobb-Douglas function at aggregate level?

*Y* = A * *K** ^{α}* *

*L*

^{(1-}

^{α)};

(4) Plot TFP (A) against GDP per capita (*y*). Is there any relationship between these two factors?